William Elwood Byerly (1849-1935) was the first student (1873) to obtain a PhD in mathematics from Harvard University. Later he returned to the University as a Professor of Mathematics where in conjunction with Benjamin Osgood Pierce, he taught the calculus course. On the basis of this teaching and the perceived needs of students, Byerly wrote Elements of Differential Calculus (1879). This was followed by Elements of Integral Calculus in 1881. The 1888 edition of Differential Calculus is examined here. In his writings on calculus, Byerly stressed the concept of limits.

On page 18 of the work, the author supplied a formal definition of the derivative of a function, which, he noted, is also a function. The notation Dx, in contemporary terms, would be considered an operator.

On page 44, Byerly defined integration as the inverse operation of differentiation.

Byerly used two methods to determine the area under a cycloid demonstrating a flexible approach to problem solving.

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